Existence and uniqueness for differential equations 02/29/16
One of the quandaries that a student of mathematics must face is proving existence and uniqueness for differential equations. Suppose we come across a differential equation dydx=f(x,y)with the initial value y(a)=b. If we want to prove that a solution exists for some xvalue, then we just have to discern if f(x,y)is continuous “near” some value (a,b) then a solution does in fact exist. Furthermore, if we would like to find if the solution is unique same near the same (a,b) value we have to take the partial derivative f(x,y)yand observe if it is continuous near (a,b)
During one’s study of Mechanical Engineering, one of the most pertinent aspects of focus is turbines. Turbines are mechanical devices that are used to collect useful energy from fluids and turn it into usable work. Usually, when a fluid becomes contingent with the blades of a turbine, then some of the kinetic energy is imparted on to the turbine, causing it to spin and thereby creating work. Turbines are used for a plethora of tasks, such as making energy through steam turbines or propelling ships such as dreadnoughts.
One of the most riveting ships in all of naval design is the dreadnought. The dreadnought uses a steam turbine driven engine to propel itself at great speeds. In addition, the dreadnought had a “big gun” design in which much of the focus of construction went into manufacturing a vast array of insurmountably powerful heavy weapons. In addition, the metallic armor made shielded the dreadnought from many potentially fatally lethal attacks. This combination of speed and power encased in a metallic fortress made the dreadnought strike fear into the hearts of all enemies. Dreadnoughts were introduced by the British navy in 1906, and were soon retired after WW1.
Electric current 02/26/16
Have you ever wondered how electricity flows? This phenomena has been labelled by Scientists and Engineers as electric current. When there is a voltage difference within a conductive wire, a flow of electrons occur between this voltage difference. The formula for current is given by the equation I=dqdt, where qis the amount of charge flowing and tis the time. There are two types of current found in Engineering, DC (direct current) and AC (alternating current). In direct current, the flow of electrons is a constant, perpetual unidirectional motion, while in Alternating current it oscillates back and forth.
Hooke’s law 02/25/16
Have you ever wondered why the force of a spring appears to grow stronger as you pull it out? This physical phenomena can be explained with the simple use of Hooke’s law. Hooke’s law states that the force of a string can be measured with the equation Fspring=k*x, with k being the spring constant and xbeing the change in distance from the resting point. The Spring constant can be found empirically by measuring the force’s change over a distance and finding the slope. we can integrate this equation in respect to x to find the potential energy of the object to obtain Uspring=12*k*x2. As one can infer, the more we stretch it out, the more potential energy is in the system, and consequently the more kinetic energy it will have when it reaches the starting point, allowing it to reach a further displacement once again.
Young’s modulus 02/24/16
Have you ever wondered why a solid body deforms when stress is applied to it? This is a consequence of Young’s modulus. To get the big picture, Young’s modulus is a property of mechanical bodies that defines how much the body deforms under stress. Before we begin, we must define the terms stress and strain. Stress is the internal forces that neighboring molecules of an continuous material apply to each other (equation is ()=FA0, Force over original area), while strain is the measure of deformation of a material (=LL0, change in length over original length). Young’s modulus is the measure of the proportion of these factors E=()which results in F*L0A0*L. The higher a bodie’s young modulus is the more resistant it is.
Wind turbines 02/23/16
Wind turbines are one of my favorite parts of Engineering. Wind turbines work by transferring energy from the wind into a spinning shaft (mechanical energy) which rotates a generator which converts all of that into electrical energy. There are two types of wind turbines, vertical axis and horizontal axis. In the latter, the larger profile allows it to catch more wind but the tower is higher and it is more complicated to build, while the former is easier to build/design/maintain but catches less energy as an offset. The part of the turbine that catches wind with blades and converts it to mechanical energy is called the rotor. One possible design for a wind turbine is known as the drag design, which uses a turbine whose axis of rotation is perpendicular is to the movement of the wind to catch the drag force provided by the wind as energy. This design is slower moving but provides a heavier torque, making it useful for tasks such as lifting water or hay 9 your stereotypical dutch farm activities) A more modern design is the lift design, which uses a rotor facing the wind movement to use the lift force caused by an asymmetric blade profile. Lift based mechanisms have a much higher rotation speed than drag types and therefore make excellent candidates for electricity generation.
The number of blades on a wind turbine can have a multitude of effects on a wind generation system. Increasing the number of blades increases efficiency at a diminishing rate. Going from one to two blades gives us a 6% increase in efficiency while going from two to three blades gives us a 3 percent efficiency. typically, the less amount of blades, the lower material and torque needed. Rotor blades are always twisted to account for a changing angle of attack, explaining the curved structure. The theoretical power generated by a wind turbine is given by the equation Power = 1/2**A*v3, with being the wind density Abeing the area of the profile andvbeing the wind velocity. The Betz limit is the theoretical efficiency of a wind turbine, given at 59.3 percent. However, most commercial systems operate at around 10-30 percent.
Wind turbines have contain a generator which converts mechanical energy into electrical energy, and transmission which increases the rotation rate of a generator to accomplish an efficient electricity generation rate. The cut in speed is the speed in which a turbine will generate usable power, typically between 7 -15 mph. The Rated speed is the speed in which a wind turbine will generate it’s designated power, typically at around 25-25 mph. To account for dangerously high wind speeds at around 45-80 mph, most wind turbines have a cut out speed feature which shuts down the wind turbine using a versatile repertory of methods. Tower heights are usually two to three times the blade diameter to account for efficiency.
Angular momentum 02/21/16
During one’s course of study of physics, one may encounter a concept known as angular momentum. Since momentum is defined as the product of the mass and movement of an object, wouldn’t it follow that there would be a special type of momentum for rotational systems, even if there was no translational movement? This quantity is known as Angular momentum. The mathematical formulation for angular momentum is given as L=I, where Iis the moment of inertia of the system and is the rotational velocity. Angular momentum can also be reformulated as L=r x p, where r is the radius, pis the linear momentum and the angular momentum is the cross product between them. Like linear momentum, Angular momentum is always conserved.
Energy Efficiency 02/21/16
As a result of the second law of thermodynamics, the phenomena of a one hundred percent energy efficient system exists in human imagination only. Consequently, the energy efficiency of systems can be measured. If you take the percentage difference between the input power of a system and the amount of work it does, then you have just measured the energy efficiency. This has a pertinent application to the field of Engineering, where maximum efficiency is paramount. For example, the energy efficiency of most commercial solar cells is between 15-20%, so 15-20 percent of the energy taken in from the sun is transferred into usable work.